Problem: If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 26$ and\[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1,\] compute $pqr$.

\begin{align*} \frac {1}{p} + \frac {1}{q} + \frac {1}{r} + \frac {360}{pqr} & = 1 \\ pq + pr + qr + 360 & =  pqr \\ 360 & =  pqr - pq - pr - qr \\  & =  (p - 1)(q - 1)(r - 1) - (p + q + r) + 1 \\  & =  (p - 1)(q - 1)(r - 1) - 25 \\ 385 & =  (p - 1)(q - 1)(r - 1) \\ \end{align*}
From here, you can factor $385$ as $5 \cdot 7 \cdot 11$, giving corresponding values of $6, 8,$ and $12$. The answer is $6 \cdot 8 \cdot 12=\boxed{576}$.